For some a,b, let f(x)=vmatrix a+( x)/(x) 1 b a 1+( x)/(x) b a 1 b+( x)/(x) vmatrix,x 0,x …

Question

For some $a,b,$ let $f(x)=\begin{vmatrix} a+\frac{\sin x}{x} \quad  1 \quad  b \ a \quad  1+\frac{\sin x}{x} \quad  b \ a \quad  1 \quad  b+\frac{\sin x}{x} \end{vmatrix},x
e 0,\underset{x	o 0}{lim}f(x)=\lambda +\mu a+
u b.$ Then $(\lambda +\mu +v{)}^{2}$ is equal to :

Accepted Answer

$\begin{gathered} \underset{x	o 0}{lim}f(x)=|\begin{array}{ccc}a+1 \quad  1 \quad  b \ a \quad  1+1 \quad  b \ a \quad  1 \quad  b+1\end{array}| \ =(a+1)(2(b+1)-b)+1(ab-a(b+1))+ba \ =(a+1)(b+2)-a+ab \ =b+a+2=\lambda +\mu a+vb \ \lambda =2,\mu =1,v=1\Rightarrow (\lambda +\mu +v{)}^{2}=16 \end{gathered}$

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